1. Field of the Invention
This invention relates to the numerical methods for simulating invicid subsonic flows and solving a class of inverse problems, which is the shape design of aerodynamic body in invicid subsonic flows.
2. Description of the Related Art
Most of the existing CFD (Computational Fluid Dynamics) works utilize the Eulerian formulation to solve the fluid flow governing equations, which means in the Eulerian plane constituted by the Cartesian coordinates the computational grid is generated in advance and mainly based on the geometry constraints. The grid forms the computational cells. Across the cell interfaces, the convective flux exists since the fluid particles pass over. It is this convective flux that causes severe numerical diffusion in the numerical solutions because the numerical diffusion is directly associated with the error resulting from numerically approximating the convective term. Since the last century, the primary CFD efforts of algorithm researchers had been concentrated on developing more robust, accurate, and efficient ways to reduce the diffusion. In particular, the upwind methods had a great deal of success in solving fluid flows, because the upwind methods reasonably represent the characteristics of the convective flux. Typically, the Godunov method[1], solving the Riemann problem on the cell interfaces, gives the most accurate results. The FVS (Flux Vector Splitting) method[2], applying the eigenstructure of the equations to treat the flux term on the cell interfaces, is more efficient. However, the convective diffusion, as a result of the fundamental nature of the Eulerian formulation, still exists in those characteristic-based numerical schemes.
On the other hand, Lagrangian description to fluid flow states the motions and properties of the given fluid particles as they travel to different locations. For the governing equations of the fluid flow, such as the Euler equations in Lagrangian formulation, there must be one coordinate representing the streamlines, which can be the stream function numerically. Another coordinate could be the distance of the particles travel. This coordinate system constitutes the Lagrangian plane, where the computational grid points are literally fluid particles and the computational grid is always simply rectangular. In particular, since the particle paths coincide with the streamlines in steady flow and no fluid particles will cross the streamlines, there is no convective flux across the cell interfaces and the numerical diffusion can be thus minimized when the equations are solved in the Lagrangian plane.
One advantage from applying the Lagrangian formulation is when solving a class of inverse problems. The typical inverse problem of aerodynamics is posed by specifying the pressure distribution on the solid-wall of aerodynamic body and determining the geometry of this solid-wall that realizes this pressure distribution. If we solve the problems in the Eulerian plane, using such as the adjoint method[3], we have to firstly estimate the unspecified body shape, then to generate a grid around this shape, and to apply the numerical methods to the flows and find the pressure distribution on the body. Next, a very important and time-consuming step is to solve the adjoint equation to modify the geometry shape. As this process has to be repeated for several successive steps for shape modification until the final target geometry is reached, the computation time of this process is usually long. A Lagrangian formulation, as mentioned before, using the stream function as one coordinate, is more suitable to work on the unspecified geometry problem, since the unspecified shape (solid-wall boundaries) are always represented by streamlines, there no need to modify the computational grid like that in the adjoint method. In the Lagrangian plane, the computational grid will keep on the same in anytime, no matter the geometry of the body shape how to change, because any solid-wall surface in Lagrangian plane is a straight line with the constant values of stream function. The method for solving the geometry shape design problem in Lagrangian plane comes to the optimal process (the most efficient process).
Although the Lagrangian formulation presents such excellent properties, it was limited in supersonic flows simulation before[4]. When the Euler equations are numerically solved in Lagrangian plane, they spatially evolve downstream, there is no any upstream information need to consider, which therefore perfectly follows the physical characteristics in supersonic flows. Recently the Lagrangian formulation had been applied on the aerodynamic shape design problems in two-dimensional supersonic flow field[5]. Until recently, for the subsonic or low speed flow domain, using the advantage of the stream function as one coordinate working on the inverse shape design problems is only for potential flows (invicid, irrotational, incompressible) and linearized compressible flows[6].
There is an apparent need to apply the advantages of the Lagrangian formulation, such as the minimum numerical diffusion in simulation and the optimal process in shape design. Some industrial applications, such as two-dimensional invicid subsonic flow simulations and design cases for nozzle and airfoil, are common, useful and necessary in the preliminary phase of product design. Numerically solving subsonic flows by using the strict Lagrangian concept becomes an excessive obstacle, physically because there exist the upstream-propagating waves in subsonic flows. Thus, the existence of a body located downstream is transmitted to the oncoming fluid particles via the waves so that the particles, sensing the influences of upstream-propagating, can change motion accordingly. A key to the success of a numerical Lagrangian method for subsonic flows lies in how to properly and instantaneously feed these upstream-propagating waves to the particles. Several objects of this invention are: (a) to provide a Lagrangian formulation of the two-dimensional Euler equations with the zeroed-out convective flux along one coordinate to minimize numerical diffusion; (b) to provide the numerical methods that solve the Euler equations in Lagrangian formulation to find more accurate solutions; and (c) to provide the numerical method that is optimal to solve the inverse shape design problems in subsonic flows.